applied sciences Article
Sizing Hybrid Energy Storage Systems for Distributed Power Systems under Multi-Time Scales Huanan Liu 1,2 ID , Dezhi Li 2 , Yuting Liu 2 Hong Zhang 1, * 1 2 3
*
ID
, Mingyu Dong 1,2 , Xiangnan Liu 3 and
Department of Electrical Engineering, Northeast Electric Power University, Jilin 132012, China;
[email protected] (H.L.);
[email protected] (M.D.) Department of Power Consumption, China Electric Power Research Institute, Beijing 100192, China;
[email protected] (D.L.);
[email protected] (Y.L.) Department of fundamental teaching, Changchun Architecture and Civilengeering College, Changchun 130022, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-432-6308-3214
Received: 15 July 2018; Accepted: 22 August 2018; Published: 24 August 2018
Abstract: With the rapid development of industry, more fossil energy is consumed to generate electricity, which increases carbon emissions and aggravates the burden of environmental protection. To reduce carbon emissions, traditional centralized power generation networks are transforming into distributed renewable generation systems. However, the deployment of distributed generation systems can affect power system economy and stability. In this paper, under different time scales, system economy, stability, carbon emissions, and renewable energy fluctuation are comprehensively considered to optimize battery and super-capacitor installation capacity for an off-grid power system. After that, based on the genetic algorithm, this paper shows the optimal system operation strategy under the condition of the theoretical best energy storage capacity. Finally, the theoretical best capacity is tested under different renewable energy volatility rates. The simulation results show that by properly sizing the storage system’s capacity, although the average daily costs of the system can increase by 10%, the system’s carbon emissions also reduce by 42%. Additionally, the system peak valley gap reduces by 23.3%, and the renewable energy output’s fluctuation range and system loss of load probability are successfully limited in an allowable range. Lastly, it has less influence on the theoretical best energy storage capacity if the renewable energy volatility rate can be limited to within 10%. Keywords: distributed power system; hybrid energy storage system; multi-time scale optimization
1. Introduction With the rapid development of the distributed photovoltaic (PV) and the wind turbine systems, power system economy and stability are facing great challenges because of the uncertainty and fluctuation of renewable energy [1–3]. To solve this problem, distributed energy storage systems are installed to suppress renewable energy outputs fluctuation, reduce the peak and valley difference of a distributed power system, improve the matching degree of daily renewable energy outputs and loads, reduce the installation capacity of distributed fossil power generators, and minimize the total costs of a distributed network [4,5]. However, due to the relatively high initial installation cost of the energy storage system, a large-scale installation of an energy storage system is difficult to achieve at present. Therefore, properly configuring energy storage capacity is crucial for improving the economy and stability of distributed power systems [6,7].
Appl. Sci. 2018, 8, 1453; doi:10.3390/app8091453
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In recent years, many models have been proposed to optimize the installation capacity of a distributed energy storage system. However, by selecting different simulation time scales and optimization criterion, the results of the optimal capacity are quite different. In [8–12], one hour is selected as the simulation time scale to configure the capacity of a distributed energy storage system. Among them, in [8], the capacity of a distributed energy storage system is configured based on the redundant energy generated by renewable energy generators. In addition, to meet the peak demands of an isolated island, Kaldellis J.K et al [9] design the optimal energy storage capacity of PV systems according to system stability constraints. Similarly, system spinning reserve and the unit commitment problem are fully considered in [10] to estimate the ideal energy storage system capacity for both grid-connected and off-grid power systems. Moreover, Lee T. et al. sized system battery capacity, aiming at reducing the average yearly energy costs of users who have participated in time-of-use (TOU) tariffs [11]. Finally, and crucially, as the mainstream method of configuring energy storage capacity, battery capacity is designed on the basis of the capability of shaving peak demands [12]. In summary, an hour-based energy storage system configuring is commonly focused on improving system economy and security, but it usually neglects the dynamic characteristic of renewable generation. To analyze the dynamic characteristic of renewable energy generation systems, one second is selected as the simulation cycle in [13] to configure the capacity of a flywheel battery hybrid energy storage system. However, Barelli L. et al. do not take system economy into account. Therefore, this paper is the first time that a battery and super-capacitor storage system is configured under multi-time scales while taking system economy, security, and the fluctuation of a renewable energy system into consideration. After translating distributed energy storage system capacity configuration problems into multi-objective optimization problems, many scholars have applied different algorithms to solve the proposed problems. Chen S.X. et al [10,14] use mixed-integer linear programming (MILP) to optimize the capacity and the operation strategy of energy storage systems for microgrids. Additionally, considering the efficiency and operation characteristics of the storage devices, dynamic programming is used in [15] to determine the optimal capacity of a vanadium redox battery (VRB). Moreover, some scholars use metaheuristic algorithms to find the optimal capacity of distributed system energy storage systems [16,17]. The aforementioned papers have good performance on translating energy storage configuration problems into multi-objective optimization problems, generating a relatively high accuracy quantitative analysis, and revealing the best capacity and operation strategy of energy storage systems. However, these papers cannot directly point out the influence of different factors on the energy storage system configuration. Therefore, this paper firstly analyzes the key factors that can affect the installation capacity of distributed energy storage systems, and afterwards intuitively displays the key factors that can affect energy storage system installation capacity in a three-dimensional (3D) graph and carries out quantitative analysis. To model renewable energy resource uncertainty and improve the accuracy of optimization results, the Monte Carlo method is extensively applied to study problems related to power system uncertainty. Liu W. et al. [18] use the pseudo-sequential Monte Carlo simulation to model the uncertainty of PV outputs, and it also combines the intelligent state space reduction to evaluate the economic performance of a power system with PV installation. In addition, based on the state sampling non-sequential Monte Carlo simulation and the DC load flow-based load curtailment model, Bakkiyaraj R.A. et al. simulated the random fault of the power system, and put forward a power system reliability evaluation method [19]. Also, by using the Monte Carlo method to simulate unbalanced voltage and then analyzing the effects of unbalanced voltage on PV hosting capacity, the hosting capacity of PV in different regions was shown in [20]. Moreover, the Monte Carlo method was used to simulate the uncertainty of wind power generation and users’ electric vehicle charging behaviors in [21]. In this paper, a joint operation model is proposed to decrease the high uncertainty of a wind power system by centralizing electric vehicle charging stations. In [18–21], the Monte Carlo method was successfully used to model the uncertainty of power systems, but this method has some limitations, such as
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for example, high complexity and low computation efficiency. In order to avoid the limitations of the Monte Carlo method, factors (excluding the uncertainty of renewable energy resource) that can affect energy storage system installation capacity are comprehensively considered when sizing hybrid energy storage systems. Then, by defining two indexes, wind speed volatility and solar irradiance volatility, the influence of renewable energy resource uncertainty on the proposed storage system is evaluated. This approach significantly reduces computation complexity. The main contributions of this article can be summarized as follows. (1) In the view of different time scales, this paper analyzes the renewable generation and load characteristics of a distributed power system, puts forward a high-precision storage capacity configuration method for a place with relatively stable meteorology, and optimizes the proposed model by using the genetic algorithm (GA). (2) By comprehensively considering the limitations of carbon emissions, costs, and the hybrid energy storage system installation capacity, this paper shows the relationship among the super-capacitor installation capacity, the battery installation capacity, and the average daily energy costs of the system in a 3D graph. Then, it generates a region of feasible hybrid energy storage system installation capacity in the 3D graph, and meanwhile reveals the influence of carbon emissions and daily energy cost on energy storage system installation capacity in a straightforward way. (3) The volatility of wind speed and solar irradiance are introduced at last to evaluate the anti-interference capability of the proposed hybrid energy storage system. 2. Modeling of Distributed Power System This section first introduces the structure of a typical distributed power system. Subsequently, the output power and carbon emissions models of all of the energy carriers in a typical distributed power system will be demonstrated in sequence. 2.1. The Structure of a Typical Distributed Power System As shown in Figure 1, a distributed power system is composed of a distributed generation system, a hybrid energy storage system, and the loads. Distributed generation mainly includes two parts: the distributed renewable energy generation and the fossil energy generation. As two typical types of renewable energy, wind power and PV are extensively installed in distributed power systems because of their environmental friendliness and easy maintenance characteristics. However, both of their outputs can be greatly influenced by the weather, which may cause network power fluctuation and bring harm to the safety operation of the power grid [22–24]. Therefore, in distributed power systems, a fossil energy generation system is usually equipped to alleviate renewable energy system output power fluctuation. In the following sections, the diesel generator will be used as an example to show the mathematical output power and carbon emissions models of a fossil energy generator equipped in the proposed power system. A distributed energy storage system, as the most important part of distributed power systems, is commonly used to shave peak demand, improve the renewable energy penetration rate, and smooth the renewable generation output power curve. However, as mentioned in the introduction, when different optimization time scales are selected, the meteorology (fluctuation characteristics) and the strong uncertainty of distributed power systems can cause different optimal results of energy storage installation capacity and operation strategy. Previous studies have preferred to take local meteorology characteristics as the key factor and select a longer time scale (for example: hour-based optimization) to configure the capacity of an energy storage system and optimize the operation of the energy storage system, which ignores the strong uncertainty (minute-based optimization) of the distributed generation [25,26]. Therefore, by taking the uncertainty and meteorology characteristics of the distributed power system into account, this paper introduces a super-capacitor and battery hybrid energy storage system to optimize the energy storage installation capacity and operation strategy. As the end of distributed power system, the loads section is also an important part of power systems. There are many ways to classify the loads, such as AC and DC loads, uncontrollable and controllable loads, etc. In this paper, in order to optimize the energy storage system configuration
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and provide feasible operation strategies, this paper only monitors the active power of the distributed power Appl. Sci.system 2018, 8, xand weakens the load classification. 4 of 17
Figure 1. The structure diagram of a distributed power system. Figure 1. The structure diagram of a distributed power system.
2.2. Modelling of Distributed Distributed Energy Energy Carriers Carriers 2.2. Modelling of 2.2.1. Distributed Renewable Power Generation System
one of of the the most most important important part part of of the the distributed distributed power power system, the As shown in Figure 1, as one wind turbine system and PV system play a critical critical role role in in reducing reducing carbon carbon emissions emissions and system system operational costs, which accelerates accelerates the pace of the installation of renewable energy generators. generators. However, the the installation of renewable generators can increase system uncertainty, which may affect However, the stable stable operation operationofofaapower powersystem. system.Therefore, Therefore, it better is better to clearly analyze factors it is to clearly analyze the the keykey factors that that can can affect the output power of renewable energy generators. affect the output power of renewable energy generators. As reported by [27], the the output output power power of of aa wind wind turbine turbine system system (P (PW W) is mainly constrained by actual wind speed (v), blade blade area area (A), (A), wind wind turbine turbine system system efficiency efficiency(η (ηW W), and air density (ρ), and its expression can can be be represented represented as as follows: follows: mathematical expression
v < v or v > v
0 v < vinin or v >outvout PW = PW = 0.5 × ρ × A ×η v33 vvin ≤ ≤ v ≤v v≤ out v out × ρ × A × ηWWv 0.5 in (
0
(1) (1)
where vvin and vout are the cut-in speed and cut-out speed of a wind turbine system, respectively. For a where in and vout are the cut-in speed and cut-out speed of a wind turbine system, respectively. For a distributed wind wind turbine turbine system, system, the the cut-in cut-in speed speed is is about about 33 m/s, m/s, and distributed and is is saturated saturated near near 10 10 m/s. m/s. As for the PV system, its output power (P PV) is strictly influenced by the ambient temperature As for the PV system, its output power (PPV ) is strictly influenced by the ambient temperature (Tamb))and equivalent solar irradiance (R), which can be expressed as [28]: (T amb and equivalent solar irradiance (R), which can be expressed as [28]: PPV
R PPV = PPV , STC × R × (1 − γ ⋅ (Ta − Tr )) = PPV,STC × RSTC× (1 − γ · ( Ta − Tr )) RSTC
(2) (2)
where:
Ta = Tamb +
R × (TNOC − 20) RSTC
(3)
In Equations (2) and (3), PPV,STC is the maximum power under the standard test condition, RSTC represents the solar irradiance level under the standard test condition, γ is a coefficient, and Tr, Ta,
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where: Ta = Tamb +
R × ( TNOC − 20) RSTC
(3)
In Equations (2) and (3), PPV,STC is the maximum power under the standard test condition, RSTC represents the solar irradiance level under the standard test condition, γ is a coefficient, and Tr , Ta , and TNOC are the reference temperature of the PV cells, the actual temperature of PV cells, and the temperature of PV cells under the normal operating condition, respectively. It can be concluded from Equations (1) and (2) that the output power of a wind turbine system and a PV system is primarily limited by the wind speed and the solar irradiance. However, the strong uncertainty of wind speed and solar irradiance can break the balance between power supply and demands, which can finally lead to power system outage. Therefore, many scholars have tried to develop complicated mathematical models to predict wind speed and solar irradiance in a more accurate way when sizing energy storage system capacity or optimizing system power flow, but this can increase the complexity of computation to some extent. To ease the process of predicting wind speed and solar irradiance, this paper first sizes hybrid energy storage system capacity by not considering the uncertainty of renewable energy. Then, it defines the maximum wind speed volatility (vv ) and solar irradiance volatility (Rv ) to represent the uncertainty of renewable generation. Finally, it uses these two variables to evaluate the anti-interference capacity of the proposed storage system (i.e., maximum acceptable renewable energy resource volatility for the designed hybrid energy storage system). In other words, this paper first sizes the energy storage system capacity, and then evaluates the influence of renewable generation uncertainty on the proposed storage system. The mathematical expressions of wind speed volatility and solar irradiance volatility are defined as follows: vv = randomk(−αw × vh , αw × vh )k (4) Rv = randomk(−αs × Rh , αs × Rh )k
(5)
In Equations (4) and (5), αw and αs are coefficients of wind speed volatility and solar irradiance volatility, respectively, while vh and Rh represent the historical wind speed and historical solar irradiance, respectively. Equations (4) and (5) show that both wind speed volatility and solar irradiance volatility have lower limits and upper limits, and that their lower limits and upper limits are directly related to the maximum volatility coefficients and the historical data of wind speed and solar irradiance. To evaluate the influence of renewable energy resource uncertainty on the proposed storage system, the modified wind speed and solar irradiance can be expressed with the summation of their historical data and their volatility values. However, it is worth noting that the modified wind speed and solar irradiance cannot be a negative value; therefore, it is necessary to limit the minimum values of the modified wind speed and solar irradiance. Equations (6) and (7) are the mathematical expressions of modified wind speed (v0 ) and solar irradiance (R0 ): v0 = max(0, vh + vv )
(6)
R0 = max(0, Rh + Rv )
(7)
Since the carbon dioxide emissions are relatively low (about 20 g/kWh) for the PV system [29] and even less for the wind turbine system, the carbon emissions of the proposed renewable energy generation system are neglected in this paper. 2.2.2. Distributed Diesel Generator System A small-scaled diesel generator system is widely deployed in distributed power systems because of its strong controllability, flexibility, high energy efficiency, and adaptability to different environments. For a distributed system, especially an off-grid system, the installation of distributed diesel generators
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can significantly reduce the system loss-of-load probability (LOLP), which will potentially improve system reliability. The mathematical expression of a diesel generator output power (Pd ) can be written as: Pd = ηd × Pdin
(8)
where η d is the energy conversion efficiency of the diesel generator, and Pdin represents the input power of the diesel generator. In this situation, the carbon emissions of the diesel generator (CO2 (Pd )) can be expressed as follows: CO2 ( Pd ) = a + b × Pd + c × Pd2 (9) where a, b, and c, the carbon emission coefficients, are respectively assigned the values of 28.144, 1.728, and 0.0017 in this paper [29]. 2.2.3. Distributed Energy Storage System There are two main types of energy storage devices: the power-type storage devices and the energy-type storage devices. Compared with the energy-type storage devices, the power-type storage devices have a faster response time and a longer lifespan [30,31]. They are widely used in renewable energy generation systems to smooth the output power of wind turbine and PV systems. The super-capacitor, one of the most mature power-type storage devices, is extensively used to stabilize power system fluctuation [32], whose maximum output power (Pscmax ) can be expressed as: Pscmax = Vsc × iscmax
(10)
Vsc is the terminal voltage of the super-capacitor, and iscmax means the maximum output current of the super-capacitor. In order to achieve its maximum output power, the capacity of the super-capacitor (Qsc ) should meet the following requirement: Qsc = where: i=
i × (dt + τ ) dV
iscmax + iscmin 2
(11) (12)
In Equations (11) and (12), i represents the average current, dV is the difference between the maximum operation voltage and the minimum operation voltage of the super-capacitor system, dt is the time of discharge, τ means the time constant, and iscmin refers to the minimum output current of the super-capacitor. Unlike the power-type storage devices, the energy-type storage devices have the advantages of longer storage time and larger energy storage capacity. Lead–acid batteries, as the most mature energy-type storage devices, are widely applied to shave the peak demand in a power system [33], whose maximum output (Pb max ) can be expressed as: Pbmax = Vb × ibmax
(13)
where Vb is the terminal voltage of the battery system, and ib max means the maximum output current of the battery. In order to improve battery lifespan, the maximum output current of the battery is limited by the installation capacity of the battery. Therefore, the initial capacity of the battery (Qb ) can be described as [33]: Qb = β × ibmax (14) where β represents the maximum discharge current coefficient of the battery. Since the distributed hybrid energy storage system (super-capacitor and battery) does not consume fossil energy and exhaust carbon dioxide, the carbon emissions of a hybrid energy storage system are ignored in this paper.
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3. System Optimization In this paper, the power system optimization problem is translated into a multi-time dimension (on the hour level and minute level) optimization problem that relates to the coordinated operation of wind power, PV, energy storage, and loads. 3.1. Objective Function This paper comprehensively considers renewable generation uncertainty, power system security, and the carbon emissions of fossil generation, and then optimizes the hybrid energy storage installation capacity with the goal of minimum generation cost. minFm = CW × EW + CPV × EPV + Cd × Ed
(15)
where EW , EPV , and Ed are the electrical energy generated by wind power, a PV system, and diesel generators, respectively. CW , CPV , and Cd represent the levelized cost of energy (LCOE) of a wind turbine system, PV system, and diesel generator system. It is worth noting that the LCOE considers various factors such as the depreciation of the system fixed assets, the operating costs of the project, the maintenance costs, and so on. Therefore, the LCOE can be used to represent the total costs of generating per-unit electricity (kWh) of different energy sources. Equation (16) is the mathematical expression of LCOE: ! n
LCOE =
∑ ( Ii + Mi + Fi )/(1 + s)i
i =0
n
/( ∑ ( Ei )/(1 + s)i )
(16)
i =0
where Ii is the investment expenditures in the year i, Mi is the operations and maintenance expenditures in i, and Fi and Ei are the fuel expenditures and electricity generation in i, respectively. s represents the discount rate, and n is the expected plant lifetime. 3.2. Constraints 3.2.1. Equality Constraints
•
Power Balance
To keep the balance of power supply and demands, the outputs of a renewable power generation system, diesel generator system, and hybrid energy storage system (PHS ) should meet the loads (PL ): PL = PW + PPV + Pd + PHS
(17)
It is worth noting that the value of PHS is positive if the hybrid energy system is discharged, and it is negative when it is charged. To keep power system security operation, the power balance constraint should always be satisfied. However, in the worst case, energy provided by the distributed generation system and the hybrid energy storage system cannot meet the loads, which makes the power system take a risk of loss of load. The mathematical expression of the loss-of-load probability (LOLP) can be written as: t LOLP = L>S (18) T In Equation (17), T is the overall simulation period, and tL>S is the loss-of-load period. 3.2.2. Inequality Constraints
•
Carbon Emission Constraint
In order to strictly control carbon dioxide emissions and reduce greenhouse gas outputs, carbon emissions need to be lower than the maximum carbon emissions (CO2max ): CO2 ( Pd ) ≤ CO2max
(19)
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Loss of Load Constraint
The increase of system loss of load probability will reduce system security. Therefore, the LOLP should be restricted to its maximum acceptable range (LOLPmax ): LOLP ≤ LOLPmax
•
(20)
Battery State-of-Charge (SOC) Constraint In order to extend the lifespan of battery system, the SOC of the battery should be set as: SOCmin ≤ SOC (t) ≤ SOCmax
(21)
where [34]:
SOC (t) =
20% min(SOC (t − 1) +
ηb ×ηc × Pbc ×τ , 90%) Qb Pbd ×τ max(SOC (t − 1) − Q ×η ×ηc , 10% ) b b
SOC (t − 1) − SDR × τ
t=0 Charging Discharging
(22)
Standby
In Equations (20) and (21), SOCmin , SOCmax , SOC(t), and SOC(t − 1) are the minimum SOC, the maximum SOC, battery SOC at time t, and battery SOC at at time t − 1, respectively. η b and η c represent the nominal cycle efficiency of the battery system and the nominal efficiency of the power conversion system. Pbc and Pbd are the charge power of the battery system measured in AC before the AC/DC conversion stage, and the discharge power of the battery system measured in the AC after the DC/AC conversion stage. τ is the dimension of the time period t, and SDR is the self-discharge of the storage system.
•
Renewable Generation Fluctuation Constraint
The fluctuation characteristics of wind speed and solar irradiance can lead to renewable system output power fluctuation. This paper takes one minute as the simulation step size, and defines that in order to keep the renewable system output power fluctuation within the certain range in a time period (∆t), the renewable power fluctuation rate (γ1 ) of the renewable system should be limited to: γ1 ≤ γmax where: γ1 =
max| PPV (t) + PW (t) − PPV (t + ∆t) − PW (t + ∆t)| PWR + PPVR
(23) (24)
In Equation (23), γmax is the maximum renewable power fluctuation rate. PW (t), PPV (t), PW (t + ∆t), and PPV (t + ∆t) represent the outputs of the wind turbine and PV systems at the time of t and t + ∆t, respectively. PWR and PPVR are the rated power of the wind turbine and PV systems.
•
Power System Peak Shaving Constraint
Similar to the wind speed and solar irradiance, power system loads also fluctuate. However, it is important to consider that a distributed power system normally contains a variety of loads, and the use of these loads can be affected by many reasons, rather than a single reason. Therefore, power system loads may not change frequently. In this situation, many papers select a longer time scale (at the hour level) to optimize the capacity of a battery storage system. At this point, the research focuses on limiting the system peak valley difference. Equation (24) is the constraint of power system peak shaving: θ1 ≤ θmax (25) where: θ1 =
PLmax − PLmin PLaver
(26)
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PL max − PL min PLaver
(26)
In Equations Equations (24) is the maximum allowance of the power system peak and valley In (24) and and (25), (25), θθmax max is the maximum allowance of the power system peak and valley difference. Meanwhile, P , P , andPPLaver arethe themaximum maximumloads, loads,minimum minimumloads, loads, and and average average Lmin Laver difference. Meanwhile, PLmax Lmax, PLmin , and are loads of of aa power loads power system. system. 3.3. Optimization Algorithm 3.3. Optimization Algorithm Since the GA is an effective algorithm to deal with single objective and multi-objective Since the GA is an effective algorithm to deal with single objective and multi-objective optimization problems, and it has clearly advantages when used to solving discontinuous optimization problems, and it has clearly advantages when used to solving discontinuous and and nonlinear problems [35], the GA is widely used to optimize power system configuration. nonlinear problems [35], the GA is widely used to optimize power system configuration. Compared Compared with other optimization algorithms, the GA can give more accurate results, although this with other optimization algorithms, the GA can give more accurate results, although this method method needs many iterations and has a relatively long computation time. needs many iterations and has a relatively long computation time. As the key step of implementing the GA, the selection of chromosomes (individuals), selection, As the key step of implementing the GA, the selection of chromosomes (individuals), selection, crossover, and mutation play a decisive role in improving the accuracy of the optimization results and crossover, and mutation play a decisive role in improving the accuracy of the optimization results accelerating the speed of computation. To balance the speed and the accuracy of calculation, this paper and accelerating the speed of computation. To balance the speed and the accuracy of calculation, this selects 20, 0.6 and 100 as the population size, crossover fraction and generations (stopping criteria), paper selects 20, 0.6 and 100 as the population size, crossover fraction and generations (stopping respectively. Figure 2 is a flow diagram to show the step of the optimizing hybrid energy storage criteria), respectively. Figure 2 is a flow diagram to show the step of the optimizing hybrid energy system installation capacity based on the proposed GA. storage system installation capacity based on the proposed GA. Start
Input system parameters (includes distributed generators, power flow and economic parameters)
Generate wind speed volatility (vv) and solar radiation volatility (Rv) in each minute, according to Equations (4) and (5)
Set the initial values of γmax and θmax
Initialize the operation parameters of the GA (population size, crossover faction and generation)
Implement the GA based on the equality constraint and inequality constraints
γmax= γmax+∆ γ, θmax= θmax+∆ θ,
Find the optimum results or reach stopping criteria No Fm
